Problem 9 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-5 i $$

Step-by-Step Solution

Answer

The polar representation of \( z = -5i \) is \( 5(\cos(-\frac{\pi}{2}) + i\sin(-\frac{\pi}{2})) \), with \( \operatorname{Re}(z) = 0 \), \( \operatorname{Im}(z) = -5 \), \( |z| = 5 \), \( \arg(z) = -\frac{\pi}{2} \), and \( \operatorname{Arg}(z) = -\frac{\pi}{2} \).

Step 1: Identify the Components of the Complex Number

The given complex number is \( z = -5i \). This number is purely imaginary with no real part. Thus, \( \operatorname{Re}(z) = 0 \) and \( \operatorname{Im}(z) = -5 \).

Step 2: Calculate the Magnitude

The magnitude (or modulus) \( |z| \) of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). Here, \( a = 0 \) and \( b = -5 \), so:\[|z| = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5.\]

Step 3: Find the Argument

The argument \( \arg(z) \) of a complex number is the angle \( \theta \) formed with the positive real axis. Since \( z = -5i \) lies on the negative imaginary axis, \( \theta \) is \(-\frac{\pi}{2}\) or \( \frac{3\pi}{2} \) depending on the convention used (either clockwise or counterclockwise reference from the positive x-axis). Both are equivalent as polar angles differ by multiple of \(2\pi\).

Step 4: Determine the Principal Argument

The principal argument \( \operatorname{Arg}(z) \) is the value of the argument constrained between \(-\pi\) and \(\pi\). Here, it is \(-\frac{\pi}{2}\), as it falls within this range by standard counterclockwise rotation from the positive x-axis.

Step 5: Represent the Complex Number in Polar Form

The polar form of a complex number is given by \( z = |z| (\cos \theta + i \sin \theta) \). Substituting the known values, we find:\[z = 5 \left( \cos\left(-\frac{\pi}{2}\right) + i \sin\left(-\frac{\pi}{2}\right) \right).\]This simplifies to the known form of \( z = -5i \) confirming the calculation since \( \cos(-\frac{\pi}{2}) = 0 \) and \( \sin(-\frac{\pi}{2}) = -1 \).

Key Concepts

complex number analysis

Complex numbers are numbers that have both real and imaginary parts. They are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with the imaginary unit \(i\) defined as \(i^2 = -1\). By analyzing complex numbers, we seek to understand these components and operations like addition, subtraction, multiplication, and division. In this exercise, the complex number given was \(z = -5i\). This is a purely imaginary number because it lacks a real component. Hence, the real part of this complex number \(\operatorname{Re}(z)\) is 0, and the imaginary part \(\operatorname{Im}(z)\) is -5. Understanding the structure and properties of complex numbers is crucial when converting them to different forms, such as polar representation.

complex number magnitude

The magnitude, also known as the modulus, of a complex number \(z = a + bi\) is the measure of its size or distance from the origin in the complex plane. It is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\). In the given exercise, the complex number \(z = -5i\) has no real part, making \(a = 0\). The imaginary part is \(b = -5\). Plugging these into the formula gives:

• \(|z| = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5\)

The magnitude of a complex number is always a non-negative real number. It helps in understanding how far the complex number lies from the point \((0,0)\) on the complex plane. This concept plays a significant role in converting a complex number from its standard form to polar coordinates.

argument of a complex number

The argument of a complex number is the angle formed between the line representing the complex number in the complex plane and the positive real axis. It is denoted as \(\arg(z)\). To determine this angle, one can use trigonometry or by recognizing the positioning on the complex plane. For the exercise \(z = -5i\), the point is directly on the negative imaginary axis.

• This locates the angle \(\theta = -\frac{\pi}{2}\) since it is 90 degrees downwards in the negative direction, or equivalently \(\frac{3\pi}{2}\) for a counterclockwise measurement.

In polar form, this angle helps describe where the number is located relative to the positive x-axis, and it is crucial for plotting or interpreting complex numbers visually. The argument can vary depending on the convention of measurement, but in analysis, ensuring it lies in the correct range is important.

imaginary and real parts of complex numbers

Understanding the imaginary and real components of complex numbers is fundamental in complex number analysis. The real part of a complex number is the coefficient of the number not associated with \(i\), while the imaginary part is the coefficient that multiplies \(i\). In the format \(a + bi\), \(a\) is the real part and \(bi\) is the imaginary part. For the complex number \(z = -5i\):

• \(\operatorname{Re}(z) = 0\) since there is no term without \(i\).
• \(\operatorname{Im}(z) = -5\) because \(b=-5\) is the coefficient of the imaginary unit \(i\).

These components provide critical information for operations involving complex numbers and are essential for transitioning between different forms, such as when expressing them in polar form where these components help to determine both magnitude and angle.